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Theorem crnggrpd 19986
Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)
Hypothesis
Ref Expression
crngringd.1 (𝜑𝑅 ∈ CRing)
Assertion
Ref Expression
crnggrpd (𝜑𝑅 ∈ Grp)

Proof of Theorem crnggrpd
StepHypRef Expression
1 crngringd.1 . . 3 (𝜑𝑅 ∈ CRing)
21crngringd 19985 . 2 (𝜑𝑅 ∈ Ring)
32ringgrpd 19981 1 (𝜑𝑅 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Grpcgrp 18756  CRingccrg 19973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364  df-ring 19974  df-cring 19975
This theorem is referenced by:  fermltlchr  32208  asclmulg  32318  ply1chr  32338  ply1fermltlchr  32339  irngss  32425  selvadd  40810  prjcrv0  41018
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